Georgia’s measure, which passed handily on Tuesday, asked voters to amend the State Constitution to allow for a commission that would approve new schools that had been rejected by local school boards.

Opponents, who said that the Constitution did not need to be amended and that charter schools already had routes of appeal, pointed to heavy spending by out-of-state donors, including Alice Walton, the daughter of the founder of Walmart, Sam Walton; Americans for Prosperity, the Tea Party group founded by the billionaire Koch brothers; and several companies that manage charter schools. Supporters of the amendment outspent opponents by about 15 to 1.

“Unfortunately, our side of the issue couldn’t be explained to the public on a bumper sticker,” said Herb Garrett, the executive director of the Georgia School Superintendents Association, which opposed the measure, saying it could divert much needed financing from traditional public schools. “That was a pretty steep hill to climb.”

In Washington, supporters of a measure to allow charter schools in the state also raised amounts that far outstripped their opponents. Donors included Ms. Walton, the Bezos foundation, and Bill Gates and Paul Allen, the co-founders of Microsoft. They raised millions of dollars to promote the ballot initiative, the fourth time in 16 years that Washington voters had been asked to approve charter schools.

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“Charter schools have cracked the code about how to reach some of these struggling populations of kids that the public schools, particularly in urban areas, are not doing well by,” said Lisa Macfarlane, Washington State director of Democrats for Education Reform.

The state teachers’ union opposes charter schools, while groups like Democrats for Education Reform and Stand for Children support them.

With about 60 percent of the votes tallied, the charter measure was leading by a slim margin, 51 percent to 49 percent. David Ammons, a spokesman for the Washington secretary of state, said that in King County, which is the state’s largest and includes Seattle, opponents were leading with less than half of the ballots counted.

Washington is one of only nine states that do not allow charter schools. The ballot measure would open the door slowly, permitting the approval of 40 schools over five years.

A version of this article appears in print on November 8, 2012, on Page P15 of the New York edition with the headline: Charter Schools Win Support In Georgia Vote. Order Reprints|Today's Paper|Subscribe